Composite Number
- Introduction
- What are Composite Numbers?
- Properties of Composite Numbers
- Checking whether a Number is a Composite or Not
- Types of Composite Numbers
- Composite Numbers from `1` to `100`
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
Factors of a number are numbers that you can multiply together to get the original number. Now, prime numbers are special because they only have two factors: `1` and the number itself. But most numbers aren't like that. They have more than two factors, and we call these numbers composite. It's like they're made up of more building blocks than primes.
What are Composite Numbers?
Composite numbers are numbers that have more than just two factors. Let’s consider `12` as an example. `12` is entirely divisible by `1`, `2`, `3`, `4`, `6`, and `12`. This means `1`, `2`, `3`, `4`, `6`, and `12` are factors of `12`. Since it has more than just two factors, we label it as a composite number.
Now let's consider the number `31`. `31` is only divisible by `1` and `31`. Since `31` has no more than two factors, we label it as a prime number.
Defining Composite Numbers
Having seen some examples of composite numbers, we can define composite numbers as all natural numbers that have more than `2` factors. A composite number is a natural positive number with two or more divisors.
Properties of Composite Numbers
`1`. A composite number is a natural number that can be divided by two or more other natural numbers.
`2`. The smallest composite number we have is `4`.
`3`. Composite numbers can be expressed as the product of two or more prime numbers.
`4`. Every even number except `2` is a composite number.
`5`. It's worth noting that `1` doesn't fall into the category of composite or prime numbers.
Checking Whether a Number is Composite or Not
How do we know if a number is a composite number or not? It's all about doing what's called a divisibility test. This involves checking if a number can be divided evenly by other numbers besides `1` and itself. We start by checking small prime numbers like `2`, `3`, `5`, `7`, `11`, and `13`. If the number we are checking is even, we start with `2`; if it ends in `0` or `5`, we check by `5`. Likewise, we can check with other smaller prime or composite numbers. If none of these factors work, we can conclude that the number is a prime number.
Let's take an example to see how it works. Let's say we have `80`. Being an even number it is divisible by `2` and as `80` ends with `0`, it is also divisible by `5`. So, it's a composite number.
Types of Composite Numbers
- Odd composite numbers
- Even composite numbers
Odd composite numbers:
These are the composite numbers that have an odd digit in the ones place. They're odd numbers that aren't prime. So, if you're thinking of numbers like `9`, `15`, and `21`, you're spot on—they're odd composite numbers.
Even composite numbers:
Now, these are the composite numbers with an even digit in the ones place. In simple terms, they're even numbers that aren't prime. Remember, excluding `2`, because it's a prime number. So, when you think of numbers like `8`, `12`, and `34`, you're thinking of even composite numbers.
Composite Numbers from `1` to `100`
Solved Examples
Example `1`. Find the smallest composite number greater than `20`.
Solution:
To find the smallest composite number greater than `20`, we can simply start checking numbers sequentially. We start with `21`, and upon checking, we find that `21` is divisible by `3` and `7`, making it a composite number. So, the smallest composite number greater than `20` is `21`.
Example `2`. Determine whether `49` is a composite number or not.
Solution:
To determine if `49` is a composite number, we need to check if it has any factors other than `1` and itself. We find that `49` is entirely divisible by `7`, `1` and `49`. Since it has more than two factors, `49` is a composite number.
Example `3`. Identify all the composite numbers between `30` and `40`.
Solution:
We can check each number between `30` and `40` to see if it has any factors other than `1` and itself. Upon checking, we find that `30`, `32`, `33`, `34`, `35`, `36`, `38`, and `39` are composite numbers within this range.
Factors of `30`: `1, 30, 2, 3, 5, 6, 10` and `15`
Factors of `32`: `1, 32, 2, 4, 8` and `16`
Factors of `33`: `1, 33, 3` and `11`
Factors of `34`: `1, 34, 2` and `17`
Factors of `35`: `1, 35, 5` and `7`
Factors of `36`: `1, 36, 2, 3, 4, 6, 9, 12` and `18`
Factors of `38`: `1, 38, 2` and `19`
Factors of `39`: `1, 39, 3` and `13`
Example `4`. Find the product of the two smallest prime numbers.
Solution:
The two smallest prime numbers are `2` and `3`. To find their product, we simply multiply them together: \(2 \times 3 = 6\). Therefore, the product of the two smallest prime numbers is `6`.
Example `5`. Determine if `100` is a composite number.
Solution:
To determine if `100` is a composite number, we check if it has any factors other than `1` and itself. Upon dividing `100` by smaller prime or composite numbers, we find that `100` is divisible by `2, 4, 5, 10, 20, 25`, and `50` in addition to `1` and `100`. Since it has more than two factors, `100` is a composite number.
Practice Problems
Q`1`. Which of the following numbers is a composite number?
- `13`
- `17`
- `21`
- `29`
Answer: c
Q`2`. What is the smallest composite number?
- `1`
- `2`
- `3`
- `4`
Answer: d
Q`3`. Which of the following numbers is a composite number?
- `37`
- `41`
- `45`
- `47`
Answer: c
Q`4`. How many composite numbers are there between `50` and `60`?
- `5`
- `6`
- `7`
- `8`
Answer: c
Q`5`. Which of the following numbers is not a composite number?
- `53`
- `57`
- `39`
- `63`
Answer: a
Frequently Asked Questions
Q`1`. What is a composite number?
Answer: A composite number is a positive integer greater than `1` that has more than two divisors or factors. In other words, it is a number that can be divided evenly by at least one number other than `1` and itself.
Q`2`. How do you determine if a number is composite?
Answer: To determine if a number is composite, you need to check if it has factors other than `1` and itself. If it does, then it is a composite number. You can perform this check by dividing the number by smaller numbers and seeing if any of them divide evenly.
Q`3`. What are some examples of composite numbers?
Answer: Examples of composite numbers include `4, 6, 8, 9, 10, 12, 14, 15, 16`, and so on. These numbers have divisors other than `1` and themselves.
Q`4`. What is the difference between prime and composite numbers?
Answer: The main difference between prime and composite numbers lies in their factors. Prime numbers have only two factors, `1` and the number itself, whereas composite numbers have more than two factors.
Q`5`. Is `1` a composite number ?
Answer: No, `1` is not a composite number. It is useful to note that `1` is also not a prime number.